I am very concerned that all of the above answers are incorrect. Or at least, they did not ask you a very important question. Did you have income while you were an F-1 student?
For the period of time when you were an F1 student, you were considered a non-resident. Your US residency begins on the date of your H1 visa if you are already in the US at that time, or the date you arrived in the US after receiving the H1 visa. If you did not have any US-sourced income while you were an F1 student, then your tax return is simple. You file a US resident tax return that only includes income beginning with your H1 visa. This can indeed be done with TurboTax.
However, if you had US income while you were an F1 student, such as a taxable fellowship, then this year you are a dual status alien. TurboTax does not handle the situation.
I believe that to file a dual-status return, you would prepare your resident return and then attach a written statement showing your nonresident taxes, and file the entire package by mail. You could potentially use TurboTax to prepare your resident return. Then if you used another tax preparation company to prepare your 1040NR nonresident return for your F1 income, that could serve as your written statement of the nonresident tax you owed. You would then put all the documents together and mail your tax return along with the payment. You can’t e-file in this situation.
it’s a complicated enough situation that you may want professional help this year. Next year when you are an H1 resident, you could easily use TurboTax.
I think @pk can confirm whether I am right or not.
